4 HAAR MEASURE ON A QUOTIENT GROUP 267 (14.4.3) (i) /// is any mapping of G into R, the number f * f(x£) rfa'(£> depends only on x = n(x). Iffe ./(G), the function xh-»J*/(*0 doc'(£) belongs to J*XG"), and we have (14.4.3.1) J/(x) da(x) = pa"(x) J */(x (ii) 7/*/w any mapping ofG into R, then (14.4.3.2) f */(x) da(x) ^ f da"(x) [ */(x{) da'({), J */ J (14.4.3.3) f /(x) da(x) £ f da"(Jc) f/(x£) da'«). J* J* J* (iii) Let N be an ^-negligible set in G, and let M ie the set of all xeG such that x~"l(N n xG') is not a' -negligible in G'. Then n(M) is a" '-negligible. The proofs follow step by step those of (13.21.3), (13.21.4), and (13.21.5), using (14.4.1) and the left invariance of a'. (14.4.4) Let u be an ^-measurable mapping of G into a topological space E, and let N be the set of all xeG such that the partial mapping % H-» w(x£) is not a! -measurable. Then 7r(N) is a" -negligible. The proof follows the lines of the proof of (1 3.21 .6) using (1 4.4.3) in place of (13.21.5). (14.4.5) Let f be an a-integrable mapping ofG into R, and let N be the set of all xeG such that the partial mapping £\-+f(x§ ofG' into R is not a'-integ- rable. Then rc(N) is VL" -negligible \ for each x <£ 7r(N), the number I f(x£) da'(£) is the same for all points x e x; the function XH~> j/(x<i;) da'(0» defined almost everywhere on G", is a"-integrable; and the formula (14.4.2) is valid. Here again, we have only to follow step by step the proof of the theorem of Lebesgue-Fubini (13.21.7), using (14.4.3) and (14.4.4). (14.4.6) Let u be an automorphism of the topological group G such that u(Gf) = G'. Let u' be the restriction ofu to G', and let u" be the automorphism ofG" induced by u (so that u"(n(x)) = n(u(x)) for all x e G). Then modG(w) = modG<(t/') • modG»(w"). balls with center 0 and contained in A,